Why are there infinitely many prime numbers?

I assume you know what a
prime number
is. There are infinitely many of them!

The following proof is one of the most famous, most often
quoted, and most beautiful proofs in all of mathematics.
Its origins date back more than 2000 years to Euclid of
Alexandria who lived around 300 BC. Euclid's argument was
different, but this is the proof that is most commonly given
today:

An interesting book on prime numbers is Paulo Ribenboim,
The New Book of Prime Number Records, 2nd ed.,
Springer Verlag, 1996, ISBN 0-387-94457-5. Starting on page
3, it gives several proofs that there are infinitely many
primes.